The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces. Application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lip...

In this paper, a distributed average tracking problem is studied for Lipschitz-type nonlinear dynamical systems. The objective is to design distributed average tracking algorithms for locally interactive agents to track the average of multiple reference signals. Here, in both the agents’ and the reference signals’ dynamics, there is a nonlinear term satisfying the Lipschitz-type condition. Thre...

In this paper, we are interested in the numerical solutions of stochastic functional differential equations (SFDEs) with jumps. Under the global Lipschitz condition, we show that the pth moment convergence of the Euler-Maruyama (EM) numerical solutions to SFDEs with jumps has order 1/p for any p ≥ 2. This is significantly different from the case of SFDEs without jumps where the order is 1/2 for...

It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If – in addition – the coefficients grow at most linearly, then this flow ha...

In this paper, we prove the existence and uniqueness result of the reflected BSDE with two continuous barriers under monotonicity and general increasing condition on y, with Lipschitz condition on z.

This paper deals with state observer design problem for a class of nonlinear dynamical systems with interval time-varying delay that satisfies the one-sided Lipschitz condition. The systems under consideration include the well-studied Lipschitz system as a special case and possess inherent advantages with respect to conservativeness. By constructing a novel Lyapunov-Krasovskii functional and ut...

We provide a Poincaré inequality for families of Lipschitz continuous vector fields satisfying a Hörmander-type condition of step two.

Iterative learning control methods are represented as powerful tools to control dynamics nowadays. Our new controller based on particular case of iterative learning control is radically different from the presented conventional method, which attempts to stabilize a class of nonlinear systems by satisfying the conditions of Lyapunov Stability Theorem. Since our algorithm is model based, its robu...

The present paper represents a continuation of paper [4], in which the existence and uniqueness problems for a general Ito–Volterra integrodifferential equation are investigated by using the concept of a non-linear random integral contractor. Since the Lipschitz condition and the random integral contractor for the coefficients of the considered equation, in general, cannot be compared, the noti...

We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting bi-Lipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximati...